Optimal Control of a Stochastic Processing System Driven by a Fractional Brownian Motion Input

Abstract

We consider a stochastic control model driven by a fractional Brownian motion. This model is a formal approximation to a queueing network with an ON-OFF input process. We study stochastic control problems associated with the long-run average cost, the infinite-horizon discounted cost, and the finite-horizon cost. In addition, we find a solution to a constrained minimization problem as an application of our solution to the long-run average cost problem. We also establish Abelian limit relationships among the value functions of the above control problems.

Stationarity and control of a tandem fluid network with fractional Brownian motion input

“…From these results, it also follows that the stationary process Z * in (4.14) and (4.15) is unique in law. Our first lemma is a variant of Proposition 4.1 of [10], and the difference here is that we allow Z 1 (0) to be a random variable.…”

“…Remark 6.1. The estimate in (6.4) also implies that lim T →∞ Z 2 (T )/T = 0 a.s. A similar estimate in [10] can be used to show that lim T →∞ Z 1 (T )/T = 0 a.s. Consequently, lim T →∞ L 1 (T )/T = u 1 a.s. Using this with (6.4) also leads to lim T →∞ L 2 (T )/T = u 1 + u 2 a.s.…”

“…Proof. Since E[Z 1 (0)] < ∞ from condition (5.3), conclusion (6.2) can be obtained by following the proof of Lemma 3.1 of [10]. For (6.3), we begin with the definition of the standard one-dimensional reflection mapping (Skorokhod map) :…”

“…Our analysis allows these FBMs to be correlated with a constant correlation coefficient. For a related one-dimensional controlled queue with FBM input, we refer the reader to [10]; our work extends that of [10] to the two-dimensional situation. The probability estimates for maximum workload of a one-dimensional queue fed by FBM were obtained in [9] and [35].…”

“…In our case, the main reason for validity of the coupling arguments is based on the uniqueness results related to the reflection map (also known as the Skorokhod map or the regulator map [13,Chapter 2.2], [30,Chapter 13.5]). A similar coupling method was used in the one-dimensional problem addressed in [10]. Our (coupling) techniques are different from those employed in [3], [8], [12], and [16].…”